The Epicycloidal Curve

The epicycloidal curve is similar to the cycloidal curve, except that
you consider the path of the point on the circle as it rotates along
the edge of a curve: the edge of another circle. Imagine a ball
rolling up a hill and down the other side. Now imagine a pinion
gear rolling around the edge of a larger gear.

If you draw a line to trace the path of the point on the circle as it
rotates along the edge of another circle, the result is called an
Epicycloidal Curve.

Since this drawing is inaccurate (for the same reasons as the
cycloidal curve drawing is inaccurate), below is a graph of the
epicycloidal curve that is much more accurate (with Rotation in Degrees on the X axis and Displacement in Inches on the Y axis).

Select this link to see the chart that was used to
create the cycloidal and epicycloidal graphs.

The addendum consists of part of the epicycloidal curve. If the
pinion has 12 leaves, the addendum consists of the first 30º
of the epicycloidal curve (because 12x30=360º). If the
pinion has 6 leaves, the addendum consists of the first 60º
of the epicycloidal curve. The other side of the addendum consists
of a mirror image of the curve. Below is a graph of an addendum using the first 60º of an epicycloidal curve. This looks like a typical clock tooth's addendum (with Rotation in Degrees on the X axis and Displacement in Inches on the Y axis).

If you select this link, you will see an animation of a gear and
lantern pinion (with 8 wires) in action. This animation was created
by Thomas Miglinci of Vienna, Austria. You can see that contact
between the gear tooth and the pinion wire does not take place until
the mid-point of the impulse, with the result that the gear does not release the
pinion wire until after the next pinion wire has entered the
disengagement phase of the impulse. This animation also
demonstrates how the curve on the addendum simulates the rolling
action desired in the interaction of the gear teeth with the pinion
wires. The gear tooth comes into contact with only a very small
portion of the pinion outside the pitch circle of the pinion (the
addendum portion), so that designing the addendum as a half circle
would be acceptable: this means that the lantern pinion is not an
inferior design. Many pinion leaves also have semi-circular
addenda.

The number of teeth on the gear determines the proportion of the
epicycloidal curve used to design the pinion leaf’s addendum, and
the number of teeth on the pinion determines the proportion of the
epicycloidal curve used to design the gear tooth’s addendum.

Therefore, the shape of a gear tooth should be different if the
pinion has 12 leaves versus 6 leaves. Since the 12 leaf pinion
occupies a much smaller angle, the shape of the gear tooth would
be based on only the steepest portion of the epicycloidal curve.
This means that the gear tooth should have a radial dedendum and
an addendum with only slight curvature and an end that follows the circumference of the addendum circle. The result
looks quite similar to a gear tooth based on the Involute Curve!